This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

Transformers are a type of neural network that have demonstrated remarkable performance across various domains, particularly in natural language processing tasks. Motivated by this success, research on the theoretical understanding of transformers has garnered significant attention. A notable example is the mathematical analysis of their approximation power, which validates the empirical expressive capability of transformers. In this study, we investigate the ability of transformers to approximate column-symmetric polynomials--an extension of symmetric polynomials that take matrices as input. Consequently, we establish an explicit relationship between the size of the transformer network and its approximation capability, leveraging the parameter efficiency of transformers and their compatibility with symmetry by focusing on the algebraic properties of symmetric polynomials.

This paper studies the binary classification of unbounded data from ${\mathbb R}^d$ generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain $\unicode{x2013}$ for the first time $\unicode{x2013}$ non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension $d$, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems.

Among the several paradigms of artificial intelligence (AI) or machine learning (ML), a remarkably successful paradigm is deep learning. Deep learning's phenomenal success has been hoped to be interpreted via fundamental research on the theory of deep learning. Accordingly, applied research on deep learning has spurred the theory of deep learning-oriented depth and breadth of developments. Inspired by such developments, we pose these fundamental questions: can we accurately approximate an arbitrary matrix-vector product using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so, can we bound the resulting approximation error? In light of these questions, we derive error bounds in Lebesgue and Sobolev norms that comprise our developed deep approximation theory. Guided by this theory, we have successfully trained deep ReLU FNNs whose test results justify our developed theory. The developed theory is also applicable for guiding and easing the training of teacher deep ReLU FNNs in view of the emerging teacher-student AI or ML paradigms that are essential for solving several AI or ML problems in wireless communications and signal processing; network science and graph signal processing; and network neuroscience and brain physics.

Deep neural networks have made significant impacts in many fields of computer science and engineering especially for large-scale and high-dimensional learning problems. Well-designed neural network architectures, efficient training algorithms, and high-performance computing technologies have made neural-network-based methods very successful in tremendous real applications. Especially in supervised learning, e.g., image classification and objective detection, the great advantages of neural-network-based methods have been demonstrated over traditional learning methods. Mathematically speaking, supervised learning is essentially a regression problem where the problem of function approximation plays a fundamental role. Understanding the approximation capacity of deep neural networks has become a key question for revealing the power of deep learning.

We study the approximation efficiency of function compositions in nonlinear approximation, especially the case when compositions are implemented using multi-layer feed-forward neural networks (FNNs) with ReLU activation functions. The central question of interest is what are the advantages of function compositions in generating dictionaries and what is the optimal implementation of function compositions via ReLU FNNs, especially in modern computing architecture. This question is answered by studying the $N$-term approximation rate, which is the decrease in error versus the number of computational nodes (neurons) in the approximant, together with parallel efficiency for the first time. First, for an arbitrary function $f$ on $[0,1]$, regardless of its smoothness and even the continuity, if $f$ can be approximated via nonlinear approximation using one-hidden-layer ReLU FNNs with an approximation rate $O(N^{-\eta})$, we quantitatively show that dictionaries with function compositions via deep ReLU FNNs can improve the approximation rate to $O(N^{-2\eta})$. Second, for H{\"o}lder continuous functions of order $\alpha$ with a uniform Lipchitz constant $\omega$ on a $d$-dimensional cube, we show that the $N$-term approximation via ReLU FNNs with two or three function compositions can achieve an approximation rate $O( N^{-2\alpha/d})$. The approximation rate can be improved to $O(L^{-2\alpha/d})$ by composing $L$ times, if $N$ is fixed and sufficiently large; but further compositions cannot achieve the approximation rate $O(N^{-\alpha L/d})$. Finally, considering the computational efficiency per training iteration in parallel computing, FNNs with $O(1)$ hidden layers are an optimal choice for approximating H{\"o}lder continuous functions if computing resources are enough.